Let $a_1, a_2, \dots$ be a sequence defined by $a_1 = a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\geq 1$. Find \[
\sum_{n=1}^\infty \frac{a_n}{4^{n+1}}.
\]
Explanation: Let $X$ denote the desired sum. Note that \begin{align*}
X &= \phantom{\frac{0}{4^0} + \frac{0}{4^1} +\text{}} \frac{1}{4^2} +
\frac{1}{4^3} + \frac{2}{4^4} + \frac{3}{4^5} + \frac{5}{4^6} +\dotsb
\\
4X &= \phantom{\frac{0}{4^0} + \text{}} \frac{1}{4^1} + \frac{1}{4^2} +
\frac{2}{4^3} + \frac{3}{4^4} + \frac{5}{4^5} + \frac{8}{4^6} +\dotsb
\\
16X&= \frac{1}{4^0} + \frac{1}{4^1} + \frac{2}{4^2} + \frac{3}{4^3} +
\frac{5}{4^4} + \frac{8}{4^5} + \frac{13}{4^6} +\dotsb
\end{align*}so that $X + 4X = 16X-1$, and $X=\boxed{\frac{1}{11}}$.